But Fulham struck three times in six minutes early in the second half to turn the match on its head.
Lewis O'Brien gave Forest hope with a 77th-minute effort but Steve Cooper's side could not salvage a point.
Forest, who lost to Bournemouth in their previous game from 2-0 up, are second-bottom of the table with four points.
Fulham, who came up from the Championship with Forest last season, moved up to sixth with 11 points.
"From the first whistle in our first game of the season, we have been on the front foot," Reed said.
"We have made it difficult against the big teams as well. The manager had some words with us at half time about where we could hurt them.
"There is a really good feeling around the club."
For Forest, back in the top flight for the first time since 1999, the alarm bells are ringing.
They have played attractive football but porous defending is undermining their season and they have conceded 14 goals in their last four defeats.
It all started well for the hosts, though, as Awoniyi headed home to reward early pressure.
Everything changed after the break as Fulham hit back in ruthless fashion with Adarabioyo's header sparking a capitulation by Forest.
Palhinha's stunning strike flattened the atmosphere in the City Ground and when Reed made it 3-1 it left Forest with too much to do despite a late flurry.
"It has happened again. A few games now we have taken the lead and not managed it well enough," Cooper said.
"We are not handling conceding the first goal. It has cost us the last two games. You cannot do that at any level, particularly this one. It is something we need to improve on."